Such a relation between \(I(a)\)  and \(\lambda (k)\)  is called a Legendre–Fenchel
transform. When \(I(a)\)  is a convex function and differentiable, or
equivalently when \(\lambda (k)\)  is differentiable, the Legendre–Fenchel
transform can be inverted and \(I(a)=\sup _{k}\left\lbrace  ka-\lambda (k)\right\rbrace  \) .
The hypothesis under which this heuristic derivation is valid are
provided by the GÃ€rtner–Ellis theorem [1]}. On domains for which the Legendre–Fenchel
is invertible and \(\lambda (k)\)  differentiable, the variational problem
(REF ) gives \(I(a)=k(a)a-\lambda (k(a))\) , where
\(k(a)\)  is given by \(a=\lambda ^{\prime }(k(a))\) .
